Dynamic investigation of hard viscoelastic materials by ball bouncing experiments


Pouyet, J.; Lataillade, J.L.

Journal of Materials Science 10(12): 2112-2116

1975


shows that forest plantations in the tropics use water according to its availability, and the appropriate

JOURNAL
OF
MATERIALS
SCIENCE
10
(1975)
2112-2116
Dynamic
investigation
of
hard
viscoelastic
materials
by
ball
bouncing
experiments
J.
POUYET,
J.
L.
LATAILLADE
Departement
des
Sciences
de
l'Ingenieur,
Laboratoire
de
Mecanique
Physique,
Uniyersite
de
Bordeaux
I,
351,
Cours
de
la
Liberation,
33405
Talence,
France
An
experimental
device
is
described
for
studying
the
behaviour
of
a
material
subjected
to
the
impact
of
a
rigid
spherical
indentor,
in
the
temperature
range
—25
to
+
90
°
C.
For
each
experimental
run
performed
on
a
quasi-semi-infinitely
hard
viscoelastic
PVC
material,
the
coefficient
of
restitution,
the
duration
of
impact
and
maximum
penetration
were
measured.
From
a
series
of
runs
performed
with
different
impact
velocities,
the
effective
damping
was
computed
as
well
as
the
dynamic
hardness.
Comparison
with
a
periodic
test
method
proved
to
be
fairly
good.
In
particular
the
same
results
on
specific
losses
were
obtained
by
both
methods,
as
long
as
the
impact
kinetic
energy
did
not
exceed
170
mJ.
Furthermore
an
estimate
of
the
storage
E
modulus
can
be
made
from
the
contact
time.
As
this
paper
deals
with
linear
viscoelasticity,
the
specific
deformation
energies
involved
are
not
too
high.
List
of
symbols
a
=
radius
of
the
contact
spherical
cap,
mm
E
l
,
E2
complex
moduli,
hbar
e
=
coefficient
of
restitution
mass
of
the
projectile,
g
R
=
radius
of
the
spherical
indentor,
mm
T
=
temperature,
°
C
T
g
=
glass
transition
temperature,
°
C
t,
=
contact
time,
msec
=
impact
velocity,
cm
sec
-1
V
R
rebound
velocity,
cm
sec
-1
x
=
penetration
depth,
mm.
µ(t)
=
relaxation
function,
hbar
tt
-1
(t)
creep
function
=
Poisson's
ratio
w
=
Angular
frequency,
rad
sec
-1
1.
Introduction
Because
synthetic
macromolecular
compounds
are
becoming
more
and
more
numerous
and
their
be-
haviour
is
different,
it
is
quite
difficult
to
define
a
single
mechanical
test
for
classifying
a
specific
compound.
Some
investigators
[1-4]
have
pointed
out
the
possibility
of
using
the
ball
bouncing
test,
particularly
for
analysing
the
2112
dynamic
hardness
of
viscoelastic
materials
[5]
.
In-
deed
because
the
strain-rates
and
impact
duration
involved
in
this
kind
of
test
are
respectively
very
high
and
brief,
resilience
and
recovery
both
play
a
part.
For
viscoelastic
materials,
the
most
satisfactory
definition
is
the
ratio
of
the
dissipative
energy
to
the
maximum
deformed
volume
AW
Pdyn
=
y
max
The
dissipation
of
energy
is
calculated
from
the
impact
velocity
and
the
coefficient
of
restitution
Ary
=
vi
2
t)
=
i
mvi
2
e
2
).
The
average
value
of
the
deformed
volume
should
be
seriously
considered,
but
experimental
difficulties
are
then
encountered.
Therefore,
the
maximum
volume
under
the
contact
cap
arbi-
trarily,
at
the
end
of
the
loading
phase
must
be
used:
V
max
=
IR
(-x-
2
)2
.
©
1975
Chapman
and
Hall
Ltd.
Printed
in
Great
Britain.
Under
these
conditions
the
dynamic
hardness
is
Pd
yn
=
2mv?
(1
_
e
2
)/71
.
Rxm
2
ax
2.
Theoretical
analysis
The
same
assumptions
as
stated
in
the
study
of
the
Hertzian
elastic
impact
[6]
,
apply,
i.e.:
a
small
rigid
indentor
(Love's
criterion)
the
sample
can
be
considered
as
a
semi-
infinite
half-space
medium
(Hunter's
criterion)
penetration
is
slight
compared
with
the
size
of
the
spherical
punch
(Hertz's
criterion)
low
impact
velocity
(Boltzmann's
criterion).
With
these
assumptions
the
problem
is
a
quasi-
steady-state
one
and
the
equation
of
motion
can
be
written
for
an
elastic
impact:
Mx"
+
8Mbt
x
312
=
0.
(2)
3(1
—v)
In
transposing
this
equation
for
the
case
of
a
visco-
elastic
material,
by
using
the
Riemann-Stieljes
integrals,
a
difficulty
arises
due
to
the
viscoelastic
behaviour
and
because
in
the
case
we
are
studying,
the
contact
radius
a(t)
reaches
a
maximum,
i.e.
the
boundary
conditions
are
not
monotonic.
Two
periods
must
then
be
distinguished:
the
loading
phase
and
the
unloading
phase.
An
important
complementary
assumption
is
used:
the
Poisson
ratio
is
assumed
to
be
constant,
which
appears
reasonable
on
the
basis
of
the
short
impact
duration
and
the
glassy
physical
state
of
the
material.
The
governing
equations
can
then
be
written
starting
from
E
q
(2)
[7]
:
(1)
for
the
loading
period
d
2
x
8
It
t<tm
dt
2
=
3M(1
v)R
J
o
1,1
and
R
x(t)
=
a
2
(t),
(4)
(2)
for
the
withdrawal
period
d
2
x
8
t,(t)
t
>
t„,
dt
2
3M(1
v)R
f
o
=
/At
t
r
)da
3
(t
I
)
and
t
d
R
x(t)
=-
a
2
(t)
r
0
(t
t
'
)
a
t
,
[L
iu
,
)
µ.(t
i
t")d(a
2
(t''))
dt'
(6)
the
t
1
(t)
function
is
defined
by
a(t
1
)
=
a(t).
A
complete
solution
of
the
impact
problem
is
available
only
through
numerical
tools.
Functions
p(t)
and
µ
-1
(t)
are
chosen
in
order
to
obtain
a
good
fit
between
numerical
and
experimental
data.
As
a
simplification,
we
can
assume
that
the
material's
behaviour
is
described
by
a
simple
Maxwell-solid
model.
This
assumption
will
be
proved
to
be
valid
for
the
material
investigated.
Let
T
be
the
relaxation
time
of
the
model.
Hence
the
relaxation
function
will
be:
n(t)
=
g
o
exp
(—
t1r),
(8)
while
the
creep
function
is
P
-1
(t)
=µo
1
(
1
+
tiT).
(
9
)
From
Equations
3
and
5
it
can
be
deduced
that
for
T:
1
e
=
9
(Or)
(10)
_
x
m
2.94
[1
+
0.195
(1
e)]
.
(11)
Then
for
a
Maxwell
solid
the
dynamic
hardness
can
be
written
using
Equations
1,
10
and
11
1
e
2
PdYn
=
rrRt
1
+
0.390
(1
e)
3.
Experimental
device
and
operating
technique
The
material
studied
was
a
Dupont
De
Nemours
PVC
and
the
indentor
was
made
of
tungsten
carbide.
The
sizes
of
indentor
and
PVC
sample
were
chosen
so
as
to
satisfy
the
theoretical
assumption
previously
stated.
The
radius
of
the
spherical
indentor
was
6.75
mm,
and
the
PVC
sample
was
cylindrical
in
shape
(thickness
40mm
and
diameter
100mm).
The
moving
indentor
was
set
on
a
steel
block
which
was
sustained
in
air
and
able
to
run
along
a
rail
(Fig.
1);
the
whole
mass
was
340
g.
This
block
was
pushed
away
at
the
start
by
the
sudden
release
of
a
coil
spring
the
com-
pression
of
which
was
altered,
by
an
electro-
magnet,
to
achieve
the
desired
impact
velocity.
The
indentor's
velocity
was
computed
from
the
time
taken
by
the
steel
block
to
run
past
a
logical
output
phototransistor
connected
to
an
intervalo-
meter.
The
PVC
sample
was
attached
to
a
heavy
concrete
mass
(about
90
kg).
t
t')da
3
(t')
(3)
(
5
)
2M
(2.94)
2
.
(12)
2113
electromagnet
spring
\
air
rail
heavy
anvil
So
that
the
plastic
sample
could
be
used
for
several
runs,
the
impact
point
was
chosen
outside
of
the
axis
of
the
sample.
The
distance
between
them
was
about
20
mm,
and
after
each
run
a
10
°
rotation
of
the
sample
was
made.
The
overall
impact
duration
was
measured
electrically.
A
thin
Cu—Ni
alloy
layer
sprayed
on
cathodically
was
placed
at
the
+
5
V
potential
of
an
electric
timer
while
the
indentor
was
connected
to
the
ground
circuit.
Penetration
depth
during
the
test
was
continuously
measured
with
a
pair
of
contactless
inductive
proximity
transducers
operating
with
100
kHz
carrier
frequency.
Measurements
were
made
with
one
of
the
transducers,
while
the
other
was
used
solely
to
adjust
the
cross
voltage.
A
variation
in
the
distance
between
the
operating
transducer
and
a
high-permeability
coil
cemented
to
the
moving
block
causes
a
change
in
the
coil's
inductivity
and
then
a
disequilibrium
of
the
Wheatstone
bridge.
A
demodulation
is
performed
which
is
recorded,
after
amplification,
on
a
storage
oscilloscope.
proximity
transducer
sample
indentor
0
projectile
Figure
1
Arrangement
of
the
experimental
device
for
the
ball
bouncing
experiment.
The
temperature
of
the
sample
is
controlled
by
the
circulation
of
a
'thermostatically
controlled
fluid
(cold
gaseous
nitrogen
or
hot
oil)
around
both
the
plastic
sample
and
its
attaching
device.
With
this
technique
the
temperature
can
be
main-
tained
in
the
range
—30
to
+
110
°
C.
4.
Experimental
results
4.1.
Contact
time
As
shown
in
Fig.
2,
for
a
given
temperature
a
log
linear
correlation
is
observed
between
V
and
t
e
,
with
the
slope
of
the
line
being
equal
to
0.20.
For
both
the
temperature
and
velocity
ranges
investi-
gated,
it
may,
therefore,
be
concluded
that
the
tested
PVC
follows
the
elastic
impact
law:
c
filsec
PVC
M.340
g
R
.6.75mm
1000
_
900_
800
_
-stec
Vi
m
sec'
0.2
0.5
1
1.5
Figure
2
Measured
contact
time
plotted
against
impact
velocity
for
five
different
temperatures
of
the
material.
2/5
t,
=
k
F
i
0.2
k
=
2.94
32N/R
[15M(1
—v
2
)
E
l
E
being
the
Young's
modulus.
(13)
Assuming
the
existence
of
an
equivalent
fre-
quency
w
for
slightly
viscoelastic
materials
(co
=
IVO,
the
dynamic
storage
modulus
E
l
(w)
can
be
defined
from
t
c
by
15M(1
P
2
)
t
c
Vi
1/5
5/2
.
(14)
E
i
(w)
=
32R
112
2.94
With
this
assumption,
a
rough
estimate
of
the
elastic
part
of
the
Young's
modulus
appears
to
be
possible
from
a
contact
time
measurement,
which
is
quite
interesting
due
to
the
simplicity
and
rapidity
of
the
measurement
[8]
.
The
isothermal
curves
plotted
in
Fig.
2
show
an
extensive
modification
of
the
variation
of
contact
time
versus
velocity
when
the
temperature
rises
to
near
T
g
=
68
°
C
which
is
the
glassy
state
transition
temperature
level.
4.2.
Coefficient
of
restitution
For
a
given
impact
velocity,
a
very
slight
variation
of
the
coefficient
of
restitution
with
temperature,
in
the
glassy
state,
is
observed
in
Fig.
3,
while
a
continuous
fall
can
be
observed
from
a
tempera-
ture
of
approximately
70
°
C:
the
PVC
then
begins
to
behave
as
a
rubber-like
material.
From
the
value
of
the
coefficient
of
restitution,
1100
700
2114
T
°
C
glass-rubber
transition
/
1.2
m
sec"'
0.9
m
sec
-
0.6
m
sec'
0.3
m
sec'
//, I/
PVC
Figure
3
Coefficients
of
restitution
as
a
function
of
the
temperature
for
four
different
impact
velocities.
0.9
0.8
0.7
-20
20
40
60
80
e,
an
estimate
is
possible
for
the
material
losses.
Indeed
the
ratio
between
the
mechanical
energy
lost
A
W,
and
the
elastic
energy
recovered,
W,
is
given
by:
OW
=
M(1'
2
ViD
1
-
e
2
(15)
WMVR
e
2
A
comparison
was
made
between
the
results
given
by
this
formula
and
those
resulting
from
a
harmonic
test
OW
=
11
tan
(16)
where
8
is
the
loss
angle
of
the
tested
material.
TABLE
I
theoretical
values
calculated
using
Equation
11
from
the
coefficient
of
restitution
and
contact
time.
As
long
as
the
temperature
is
lower
than
T
g
,
very
good
agreement
is
observed
between
the
two
sets
of
values.
However
a
small
shift
is
ob-
served
between
the
two
sets
of
results
(Fig.
4).
For
determining
distances
between
the
high-p
coil
on
the
moving
block
and
the
transducer,
it
was
not
possible
to
obtain
the
exact
relative
initial
position
between
coil
and
transducer
for
both
the
standardization
procedure
and
the
experimental
runs.
Hence
a
determination
error
exists
which
is
constant
as
long
as
temperature
and
sample
are
not
changed.
PVC
C)
V
l
(m
sec
-I
)
0.45
0.56
0.35
0.52
0.37
0.46
1
-
e
2
11
tan
6
e
2
0.13
0.15
0.13
0.16
0.33
0.40
0.15
0.17
0.22
20
56
70
x
mox
,„„,
Te.T
g
OA
0.3
0.2
0.1
Ta
T
g
0.4
0.6
Table
I
shows
good
agreement
between
the
two
ways
of
determining
the
damping
capacity
tan
(5)
for
the
glassy
state
as
long
as
the
impact
velocity
does
not
exceed
0.5
m
sec
-1
.
This
result
appears
quite
useful
because
of
the
great
ease
in
recording
the
results
of
an
impact
test.
4.3.
Maximum
penetration
depth
and
dynamic
hardness
A
comparison
was
made
between
the
experimental
values
for
the
maximum
penetration
depth
and
I..
0.8
0.4
0.6
0.8
o
measured
values
computed
values
for
o
Maxwellian
solid
(from
Eq.11)
Figure
4
Maximum
penetration
depth
plotted
against
Vi
X
t
e
,
below
and
up
above
the
glass
temperature
level.
For
a
given
temperature,
all
the
runs
were
per-
formed
with
the
same
sample,
which
was
only
turned
round
on
its
axis
after
each
run.
The
error
is,
therefore,
the
same
for
a
series
of
experiments.
In
spite
of
the
experimental
error
just
de-
scribed,
the
experimental
results
enable
us
to
con-
2115
Pd
yn
hbar
15
10
PVC
W
k
.1
0.07
0.14
0.21
Figure
5
Values
of
the
dynamic
hardness
plotted
against
the
kinetic
energy
of
the
projectile.
elude
that
the
mechanical
behaviour
of
the
PVC
samples
studied
can
be
described
by
a
simple
Maxwell
model,
but
only
for
the
glassy
state.
equation
(12)
can,
therefore,
be
used
to
estimate
the
dynamic
hardness
as
a
function
of
the
kinetic
impact
energy
(Fig.
5).
For
the
impact
velocity
range
explored
the
dynamic
hardness
of
PVC
was
found,
in
the
glassy
state,
to
be
independent
of
temperatures.
When
the
PVC
exhibits
a
rubber-like
behaviour,
i.e.
for
T>T
g
,
Equation
1
has
to
be
used
directly.
Experimental
determination
of
penetration
depth
must
then
be
as
thorough
as
possible
for
the
effective
determination
of
dynamic
hardness.
5.
Conclusion
In
conclusion
it
must
be
emphasized
that
the
im-
pact
test
can
be
used
as
an
efficient
and
quick
method
to
experimentally
evaluate
the
mechanical
characteristics
of
linear
viscoelastic
materials:
i.e.
(1)
damping
capacity,
(2)
dynamic
storage
modulus
and
(3)
dynamic
hardness.
In
the
case
of
materials
obeying
a
Maxwell
model,
the
determination
of
dynamic
hardness
is
straightforward
from
two
sets
of
experimental
data:
i.e.
the
contact
time
and
the
coefficient
of
restitution.
References
1.
H. H.
CALVIT,
J.
Mech.
Phys.
Solids
15
(1967)
141.
2.
Y.
H.
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